#6495040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,23,24)(19,21,26,22)(20,25,30,32)(27,31,28,29);(9,10)(11,14)(12,13)(15,16)(17,19,22,18,23,26,21,24)(20,28,29,32,30,27,31,25);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(18,24)(19,26)(25,32)(27,28);(10,13)(14,16)(17,20)(18,25)(19,27)(21,31)(22,29)(23,30)(24,32)(26,28);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,19,24,26)(20,29,30,31)(25,28,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402236);